Showing papers on "Riemann curvature tensor published in 1975"

Homogeneous riemannian spaces of negative curvature

Abstract: In this paper the structure of simply-transitive isometry groups of a Riemannian space with nonpositive curvature is studied. The results obtained are then applied to classify Riemannian spaces of negative curvature possessing a metabelian transitive isometry group and also to classify homogeneous Einstein spaces with nonpositive curvature and with dimension .Bibliography: 13 items.

Singularities in globally hyperbolic space-time

Abstract: A singularity reached on a timelike curve in a globally hyperbolic space-time must be a point at which the Riemann tensor becomes infinite (as a curvature or intermediate singularity) or is of typeD and electrovac.

Some restrictions on algebraically general vacuum metrics

Abstract: Vacuum Einstein metrics of Petrov type I, general, are considered. It is shown that the only solution of this sort in which one of the Petrov scalars is zero is the trivial flat−space one. Further, it is shown that the point at which the four Petrov scalars vanish simultaneously (zero curvature tensor) cannot be included as a regular point of a neighborhood over which the scalars are functionally independent. In fact, for type I all derivatives of the Petrov scalars must vanish at a point at which the curvature tensor does so that this point cannot be a regular point of any nontrivial analytic solution.

Structural equations for Killing tensors of order two. I

Abstract: For any Killing tensor Kαβ of order two, a system of linear homogeneous first−order differential equations of the form (FA);α = JB(ΓαABFB) is derived. F1, F2, ... are the components of the tensors Kαβ, Lαβγ = 2Kγ[β;α], and Mαβγδ = (1/2)(Lαβ[γ;δ] + Lγδ[α;β]). The coefficients ΓαAB are linear expressions in the Riemann tensor and its covariant derivative. These equations are analogous to those satisfied by a Killing vector Kα and the Killing bivector ωαβ = Kβ;α, with Lαβγ and Mαβγδ playing roles analogous to ωαβ. The tensor Lαβγ has the symmetries Lαβγ = − Lβαγ and L[αβγ] = 0, and Mαβγδ has the symmetries of the Riemann tensor. Several relations similar to those satisfied by covariant derivatives of Killing vectors are derived. Perspectives for further work are briefly discussed with the idea of using the equations to investigate space−times which admit Killing tensors of order two.

On the curvature dynamics for metric gravitational theories

Abstract: Any ’’metric gravitation theory’’ (including general relativity) is shown to determine transport equations for the connection and curvature of the Lorentz frame bundle P4 defined by the metric g. Observers are generally defined as curves in P4 which project down to timelike trajectories in space–time. The transport of curvature along an observer trajectory is then given by a Lorentz Lie algebra‐valued current composed of an internal and external part. Einstein’s equations are shown to define one part of the self‐dual limit of the usual Yang–Mills gauge equations, here called a particular form of curvature dynamics. As a consequence, the Yang–Mills‐like energy–momentum tensor, introduced for the Lorentz connection, vanishes identically under Einstein’s vacuum conditions.

The uniqueness of g ij in terms of R { /ijk l }

Abstract: Some theorems are given which show when a curvature tensor is the curvature of pseudometric and when this pseudo-metric is unique. These results not only contribute to the exact solution work done previously but also throw some light on Mach's principle in general relativity.

On vacuum metrics of type (3, 1)

Abstract: A technique is described for constructing solutions of Einstein's equations for empty space, in which the Riemann tensor has a triply degenerate principal null direction with twist.

Instability of intermediate singularities in general relativity

Abstract: It has been shown that "intermediate" singularities, where all Riemann tensor invariants are finite, occur in certain cosmological models. Associated with the singularities are Cauchy horizons, across which the matter flows into a stationary region of space-time. We investigate scalar-wave propagation in these spaces. Our results suggest that the intermediate singularities become localized curvature singularities while the Cauchy horizons are a stable feature of the models.

Group theory of the massless spin 2 field and gravitation

Abstract: The simplest manifestly covariant unitary representation of the Poincare group for zero mass and spin 2 is constructed. This representation is carried by fourth rank tensors which satisfy the equations of the Riemann curvature tensor in the linearized theory of gravitation in vacuo. In particular, the requirement of unitarity implies the Bianchi identities.

Exact solution for the gravitational field of a charged, magnetized, spinning mass

Abstract: A family of solutions of the Einstein-Maxwell field equations is presented, corresponding to the exterior of stationaryaxisymmetric sources with charge, mass, angular momentum, and magnetic dipole moment. The Riemann tensor vanishes asymptotically for each member of the family; some solutions are asymptotically flat and some have NUT-like behavior asymptotically. For the asymptotically flat solutions, the gyromagnetic ratio may vary from zero to one. The corresponding value for the Kerr-Newman solution is one. A method for generating infinite chains of families of solutions of the Einstein-Maxwell equations is described.

Geometrical description of field algebra

Abstract: The algebraic presentation, of the curvature tensor suggested earlier in [1] yields a simple form of the energymomentum tensor, presupposed by the Sugawara model. This expression gives rise to the expression of the Yang-Mills fields in terms of tetrads. This provides for an interpretation of the Yang-Mills fields as classical counter-parts of inhomogeneous group generators. A generalized form for the Yang-Mills fields is obtained for which commutation relations of the field algebra follow naturally.